Theorem 4.1 
In the sense of [[1]}], the map \(\widetilde{\Phi }\)  is a perfect relative obstruction theory for \(\widetilde{\mathbf {M}}^s\)  over \(S\) . And there exists a virtual fundamental class \([\widetilde{\mathbf {M}}^s]^{\mathrm {vir}}\in A_{\mathrm {vdim}+\dim S}(\widetilde{\mathbf {M}}^s)\)  of virtual dimension \(\mathrm {vdim}=\mathrm {rk}(\widetilde{\mathbf {E}}^\bullet )\) .

